OID(Object Identifier) denotes an object.
Examples:
------------------------------------------------------------------
OID                                      object
------------------------------------------------------------------
1.3.14.3.2.26                       SHA-1
2.16.840.1.101.3.4.2.1        SHA-256
1.2.840.113549.1.7.2          PKCS-7 signedData
------------------------------------------------------------------

In OpenSSL no functions are directly provided to compute the OID ASN.1 encode. At lease two methods can be taken into account.

1. Create a temporary object by invoking function OBJ_create(), then encode it by invoking function i2d_ASN1_OBJECT().
Implementation (Not recommended)

/**************************************************
* Author: HAN Wei
* Author's blog: http://blog.csdn.net/henter/
* Date: Oct 11th, 2014
* Description: implement the OID ASN.1 encode function
**************************************************/

#include <stdio.h>
#include <openssl/objects.h>
#include <openssl/asn1.h>

int Asn1EncodeOid(char *oid,
                  unsigned char *encode,
                  int *encode_len)
{
  int new_nid, byte_len;
  ASN1_OBJECT *obj;
  unsigned char *tmp_pointer;

  new_nid = OBJ_create(oid, "oid example", "Object Identifier Example");
  obj = OBJ_nid2obj(new_nid);

  if (!encode)
  {
    byte_len = i2d_ASN1_OBJECT(obj, NULL);
	if (byte_len <= 0)
	{
#ifdef _DEBUG
      printf("get ASN.1 encode byte length failed at %s, line %d!\n", __FILE__, __LINE__);
#endif
	  OBJ_cleanup();
	  return (-1);
	}
	else
	{
	  *encode_len = byte_len;
	  OBJ_cleanup();
	  return 0;
	}
  }
  else
  {
    tmp_pointer = encode;
    byte_len = i2d_ASN1_OBJECT(obj, &tmp_pointer);
    if (byte_len <= 0)
    {
#ifdef _DEBUG
      printf("ASN.1 encode OID failed at %s, line %d!\n", __FILE__, __LINE__);
#endif
      OBJ_cleanup();
      return (-1);
    }
    else
    {
      *encode_len = byte_len;
      OBJ_cleanup();
      return 0;
    }
  }
}

This is not a good implementation. OBJ_cleanup() will free all dynamically created object, so this function must be used carefully. Especially when multiple threads are running, the fact that one thread invokes OBJ_cleanup() may run the risk of cleaning object created by other threads. The consequence is unpredictable.

2. Compute OID payload part ASN.1 encode by invoking function a2d_ASN1_OBJECT() firstly, compute the OID encode by invoking function i2d_ASN1_OBJECT() next.

A complete Implementation (recommended)

Header file:

/**************************************************
* File name: oid_encode.h
* Author: HAN Wei
* Author's blog: http://blog.csdn.net/henter/
* Date: Oct 11th, 2014
* Description: declare the OID ASN.1 encode function
**************************************************/

#ifndef HEADER_OID_ASN1_ENCODE_H
  #define HEADER_OID_ASN1_ENCODE_H

#ifdef  __cplusplus
extern "C" {
#endif

/**************************************************
* Name: Asn1EncodeOid
* Function: compute ASN.1 encode for a specific OID
* Parameters:
    oid        [in]   OID string terminated with '\0'
    encode     [in]   buffer used to store OID ASN.1 encode
	encode_len [out]  byte length of OID ASN.1 encode
* Return value:
   succeed -- 0
   fail    -- -1
* Notes:
1. If the NULL pointer is assigned to parameter 'encode',
   this function does not perform ASN.1 encode. The OID ASN.1
   encode length is assigned to parameter 'encode_len' and
   the function returns.
2. If the value assigned to parameter 'encode' is not NULL,
   the OID ASN.1 encode is written into the buffer pointed by
   parameter 'encode', and encode length is assigned to
   parameter 'encode_len'. In this case the buffer length is
   NOT checked before the encode is written into the buffer.
   MAKE SURE that the buffer length is big enough to accomodate
   the ASN.1 encode!
**************************************************/
int Asn1EncodeOid(char *oid, unsigned char *encode, int *encode_len);

#ifdef  __cplusplus
}
#endif

#endif /* end of HEADER_OID_ASN1_ENCODE_H */

Function implementation file:

/**************************************************
* File name: oid_encode.c
* Author: HAN Wei
* Author's blog: http://blog.csdn.net/henter/
* Date: Oct 11th, 2014
* Description: implement the OID ASN.1 encode function
**************************************************/

#include <stdio.h>
#include <openssl/objects.h>
#include <openssl/asn1.h>

int Asn1EncodeOid(char *oid,
                  unsigned char *encode,
				  int *encode_len)
{
  int payload_len, total_len;
  ASN1_OBJECT obj;
  unsigned char *tmp_pointer, *payload_encode;

// get payload ASN.1 encode
  payload_len = a2d_ASN1_OBJECT(NULL, 0, oid, -1);
  if (payload_len <= 0)
  {
#ifdef _DEBUG
    printf("get ASN.1 encode byte length failed at %s, line %d!\n", __FILE__, __LINE__);
#endif
    return (-1);
  }
  if ( !(payload_encode=(unsigned char *)malloc(payload_len)) )
  {
#ifdef _DEBUG
    printf("invoke malloc() function failed at %s, line %d!\n", __FILE__, __LINE__);
#endif
    return (-1);
  }
  payload_len = a2d_ASN1_OBJECT(payload_encode, payload_len, oid, -1);
  if (payload_len <= 0)
  {
#ifdef _DEBUG
    printf("ASN.1 encode payload failed at %s, line %d!\n", __FILE__, __LINE__);
#endif
    free(payload_encode);
    return (-1);
  }

// get the whole OID ASN.1 encode
  obj.data = http://www.mamicode.com/payload_encode;>

A demo program file:
/**************************************************
* File name: test.c
* Author: HAN Wei
* Author's blog: http://blog.csdn.net/henter/
* Date: Oct 11th, 2014
* Description: this program demonstrates how to invoke
    the OID ASN.1 encode function
**************************************************/

#include "oid_encode.h"
#include <stdio.h>
#include <stdlib.h>
int main(void)
{
  char oid[128]="2.16.840.1.101.3.4.2.1";  /* SHA-256 OID*/
  unsigned char *buffer;
  int buffer_len, i;

  if ( Asn1EncodeOid(oid, NULL, &buffer_len) )
  {
    printf("error detected!\n");
#if defined(_WIN32) || defined(_WIN64)
    system("pause");
#endif
    return (-1);
  }
  printf("OID ASN.1 encode length is %d bytes.\n", buffer_len);

  if ( !(buffer = (unsigned char *)malloc(buffer_len)) )
  {
    printf("invoke malloc() function failed!\n");
#if defined(_WIN32) || defined(_WIN64)
    system("pause");
#endif
    return (-1);
  }

  if ( Asn1EncodeOid(oid, buffer, &buffer_len) )
  {
    printf("error detected!\n");
    free(buffer);
#if defined(_WIN32) || defined(_WIN64)
    system("pause");
#endif
    return (-1);
  }
  printf("OID ASN.1 encode:\n");
  for (i=0; i<buffer_len; i++)
    printf("0x%x  ", buffer[i]);
  printf("\n");

  free(buffer);
#if defined(_WIN32) || defined(_WIN64)
  system("pause");
#endif
  return 0;  
}
ASN.1 encode of SHA-256 is obtained from the demo:  0x6 0x9 0x60 0x86 0x48 0x1 0x65 0x3 0x4 0x2 0x1
This is a better implementation.
ASN.1 Encode an Object Identifier (OID) with OpenSSL